The smallest mammal is a type of shrew which weighs about 1.5 grams. The smallest bird seems to be
the hummingbird which weighs about the same as the smallest shrew. There are smaller fish and insects but
this seems to be a lower limit
for land creatures made of flesh and blood. The largest land mammal is the elephant, for which a typical
weight is five thousand kilograms. This is a ratio of over three million to one. The question is what factors determine that lower limit and upper limit
of animal size.

Analysis

There is a strong possibility that the answer lies in the relationship of surface area and volume.
The scale of a creature is some linear dimension such height or length.
Let s be the scale of animal of a particular shape, say quadruped. The need for food to satisfy
the metabolism of the
cells is proportional to the volume of the animal and thus proportional to the cube of the scale, say
αs³. The need for food to manintain at constant temperature, the thermal need, is proportional to
the surface area of the animal and thus to the square of the scale, say βs². The need for food
is the maximum of those two needs in that the metabolism of the cells also provide heat for maintaining
body temperature.

When considering the food that an animal may obtain it is most convenient to let that be the net food
energy obtained; i.e., the energy used up in finding, capturing, ingesting and digesting is deducted.
The maximum amount of food obtained is a function of the area that the animal can forage over in a single
day. This is proportional
to the distance the animal can cover in a foraging day times the width of the foraging path. This is equal
to stride length times the stride
frequency times the width of the forage path. The stride frequency is a function of the pendulum period
of an animal leg which is inversely proportional
to the square root of leg length. For more on this topic see Animal Running
The leg length is proportion to scale and the width of the foraging path is also proportional to scale.
Thus the food which can be
foraged is proportional to the three halves power (1+1-½=3/2) of scale; say γs3/2.
This is what animals could get as maximum yield from foraging, not what they actually get. Generally they
do not have to get this maximum.

The feasible animal scales are those for which

γs3/2 ≥ max[αs3, βs2]

For large scale animals the cubic term is dominant. Thus the largest scale animal would be the
one for which

γsmax3/2 = αsmax3and thus
smax3/2 = (γ/α)
and hence
smax = (γ/α)2/3

The diagram below illustrates the concepts.

The model indicates that the largest animals would be of a scale such that lumbering about as fast as they
can they can just barely find enough to eat to satisfy their metabolic needs.

At the other end of the scale spectrum the thermal requirement is thought to dominate.
However with the thermal needs being proportional to the second power of scale but the maximum foraging
yield depending upon the 3/2 power of scale there is no way the curves can properly define a minimum scale.
If the thermal need is greater than the foraging yield for very small scales this will prevail at all scales
because of the size of the exponents. The foraging yield curve and the thermal need curve can cross but only
such as to define a maximum scale size not a minimum one. The diagrams below illustrates this.

In this diagram the green foraging function curve crosses the red thermal needs line from above, indicating
that scales below that level can survive but not above the crossing point.

In this diagram the green foraging function line is always below the red thermal needs line.

In order for there to be both a lower limit and upper limit defined the exponent of the foraging yield function has to be
has to be between the 2 of the thermal need function and the 3 of the metabolic need function.

The analysis above only included the search and capture time in the foraging yield. There is also the
time required to ingest the food. Let the size of the food parcel (prey or plant) be μs³. This is
based upon the observation that bigger animals have bigger prey, smaller animals smaller prey. The running
speed is proportional to the square root of scale, say ωs½. Let the distance between
prey be δs. The time to get from one prey to the next is then δs/ωs*frac12; or let
say θs½.

The time required for an animal to get the prey down its gullet is the time requred to
get something of volume proportional to s³ down an orifice of area proportional to s²; i.e. a time
proportional to s, say εs. Thus the total time between prey is θs½+εs.
This divided into the unit of time is the number prey ingested per unit time. This multiplied by the size
of the prey μs³ is the food obtained per day. This is

μs³/[θs&fract12; + εs]
which for large s works out to be approximately proportional to s² but for small s is roughly proportional
to s5/2. This allows for both an upper and lower limit to animal scale to be defined.

The diagram below illustrates a case for this type of foraging function.

Where the green foraging function curve crosses the red thermal needs curve is the mininum animal scale.
Below that level an animal cannot get enough to eat to keep from becoming hypothermic. Where the green foraging
function curve crosses the blue metabolic needs line is the maximum scale. Above that level the animal
cannot get enough food to meet its metabolic needs. Between the minimum scale and the maximum scale the
foraging curve is above both the thermal and metabolic needs curves so animals of those scales are feasible.

For the functions defined above the minimum scale smin is defined as a solution to

μs³/[θs&fract12; + εs] = βs²
or, equivalently
μs³ = θβs5/2 + εβs³

This can be solved as a sixth power polynomial in √s.

Of course the minimum scale might be determined by some other factor than the matter of heat loss. The fact
that shrew live in high latitude regions supports this possibility.

The heart beat of creatures necessarily increases with smaller size. There might be
some upper limit to the heart rate. Both shrews
and hummingbirds have heart rates that reach 1200 beats per minute under measurement conditions.
The average could be much less than half that level, say 500 beats per minute because
clearly the heart rate during sleep is much less than that for a creature flying or scurrying about.
This is in contrast to the human heart rate of
about 60 beats per minute and that of an elephant of 30 beats per minute. Heart tissue may not be able to
contract and relax faster than 1200 times per minute.

Hearts regardless of scale last about one billion beats on average. At a rate of 500 beats per minute
that billion limit is used up in about 3.8 years. The actual average life span of
hummingbirds in the wild is three to four years.

There could be the lower limit for mammals and birds to go through
mating, gestation or incubation and raising their offspring to self sufficiency. But with a high mortality
rate a pair might have to, on average, go through more than one season to ensure having two offspring reach
the age of reproduction.